Mathematics exemplar problems class 10 Solutions for Chapter 8 introduction to trigonometry and its equations

Given that sinθ =\frac ab, then cosθ is equal to

The value of (tan1° tan2° tan3° … tan89°) is

If ΔABC is right angled at C, then the value of cos (A + B) is

If sin A = ½ , then the value of cot A is

If sin A + sin² A = 1, then the value of the expression (cos² A + cos⁴ A) is

Given that sin α = 1/2 , cos β = 1/2 , then value of (α + β) is

The value of the expression \left[\frac{\sin^222^{\circ}+\sin^268^{\circ}}{\cos^222^{\circ}+\cos^268^{\circ}}+\sin^263^{\circ}+\cos63^{\circ}\sin27^{\circ}\right] is

If cos (α + β) = 0, then sin (α – β) can be reduced to

If cos 9α = sinα and 9α < 90°, then the value of tan5α is

sin(45° + θ) – cos (45° – θ) is equal to

If 4 tan θ = 3, then \left[\frac{4\sin\theta-\cos\theta}{4\sin\theta+\cos\theta}\right] is equal to

If cos A = 4/5, then the value of tan A is

If sin θ – cos θ = 0, then the value of (sin⁴ θ + cos⁴ θ) is

The value of the expression [\operatorname{cosec}(75°+θ)-\sec(15°-θ)-\tan(55°+θ)+\cot(35°-θ)] is

State whether the following statment is true or false.

tan 47^o/cot 43 ° = 1

State whether the following statment is true or false.

\sqrt{\left(1-\cos^2\theta\right)\sec^2\theta}=\tan\theta

State whether the following statment is true or false.

The value of the expression (sin 80° – cos 80°) is negative.

State whether the following statment is true or false.

If cosA + cos²A = 1, then sin²A + sin⁴A = 1.

State whether the following statment is true or false.

The value of the expression (cos²23° – sin²67°) is positive.

State whether the following statment is true or false.

If the height of a tower and the distance of the point of observation from its foot, both are increased by 10%, then the angle of elevation of its top remains: unchanged.

State whether the following statment is true or false.

(tan θ + 2) (2 tan θ + 1) = 5 tan θ + sec² θ.

State whether the following statment is true or false.

If a man standing on a platform 3 m above the surface of a lake observes a cloud and its reflection in the lake, then the angle of elevation of the cloud is equal to the angle of depression of its reflection.

State whether the following statment is true or false.

The value of 2 sinθ can be \left(a+\frac{1}{a}\right), where a is a positive number, and a ≠ 1

State whether the following statment is true or false.

The angle of elevation of the top of a tower is 30°. If the height of the tower is doubled, then the angle of elevation of its top will also be doubled.

State whether the following statment is true or false.

\cos \theta=\frac{a^{2}+b^{2}}{2 a b}, where a and b are two distinct numbers such that ab > 0

If the length of the shadow of a tower is increasing, then the angle of elevation of the sun .............

State whether the following statment is true or false.

If tan A = ¾, then sinA cosA = \frac{12}{25}

State whether the following statment is true or false.

(√3+1) (3 – cot 30°) = tan³ 60° – 2 sin 60°

State whether the following statment is true or false.

1+\frac{\cot ^{2} \alpha}{(1+\operatorname{cosec} \alpha)}=\operatorname{cosec} \alpha

A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, find the height of the wall.

Find the angle of elevation when the shadow of a pole 'h' m high is h\sqrt{3 } m long.

Express tan⁴ θ + tan² θ in terms of sec θ

If \sqrt{3}\tanθ=1, then find the value of sin² θ – cos² θ.

Simplify: (1 + tan² θ) (1 – sin θ) (1 + sin θ)

If 2 sin² θ – cos² θ = 2, then find the value of θ.

An observer 1.5 m tall is 20.5 m away from a tower 22 m high. Determine the angle of elevation of the top of the tower from the eye of the observer.

State whether the following statment is true or false.

\frac{\cos ^{2}\left(45^{\circ}+\theta\right)+\cos ^{2}\left(45^{\circ}-\theta\right)}{\tan \left(60^{\circ}+\theta\right) \tan \left(30^{\circ}-\theta\right)}=\mathbf{1}

State whether the following statment is true or false.

\frac{\sin\theta}{(1+\cos\theta)}+\frac{(1+\cos\theta)}{\sin\theta}=2\operatorname{cosec}\theta

State whether the following statment is true or false.

tan θ + tan (90° – θ) = sec θ sec (90° – θ)

State whether the following statment is true or false.

\frac{\tan\mathrm{A}}{1+\sec\mathrm{A}}-\frac{\tan\mathrm{A}}{1-\sec\mathrm{A}}=2\operatorname{cosec}\mathrm{A}

State whether the following statment is true or false.

(sin α + cos α) (tan α + cot α) = sec α + cosec α

The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is 60°, and the angle of elevation of the top of second tower from the foot of first tower is 30°. Find the height of the second tower.

From the top of a tower h m high, the angles of depression of two objects, which are in line with the foot of the tower are α and β, (β > α). Find the distance between the two objects.

A ladder rests against a vertical wall at an inclination α to the horizontal. Its foot is pulled away from the wall through a distance p, so that its upper end slides a distance q down the wall and then the ladder makes an angle β with horizontal. State whether if LHS equals to RHS for \frac{p}{q}=\frac{\cot\beta-\cos\alpha}{\sin\alpha-\sin\beta}.

A window of a house is h m above the ground. From the window, the angles of elevation and depression of the top and the bottom of another house situated on the opposite side of the lane are found to be α and β respectively. Find the height of the other house.

The angle of elevation of the top of a tower from certain point is 30°. If the observer moves 20 metres towards the tower, the angle of elevation of the top increases by 15°. Find the height of the tower.

The angle of elevation of the top of a tower from two points distant s and t from its foot are complementary. The height of the tower is

The shadow of a tower standing on a level plane is found to be 50 m longer when Sun’s elevation is 30° than when it is 60°. Find the height of the tower.

If sin θ + cos θ = p and sec θ + cosec θ = q, check if q(p² – 1) = 2p.

State whether LHS equals to RHS :

\frac{1+\sec\theta-\tan\theta}{1+\sec\theta+\tan\theta}=\frac{1-\sin\theta}{\cos\theta}

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height ‘h’. At a point on the plane, the angles of elevation on the bottom and top of the flag staff are α and β, respectively. The height of the tower is

If tanθ + secθ = l, check if \sec \theta=\frac{1^{2}+1}{21}

If a sin θ + b cos θ = c, then state whether a\cos\theta-{b}\sin\theta=\sqrt{a^2+b^2-c^2}

If 1 + sin²θ = 3sinθ cosθ , then find the value of tanθ.

Given that sinθ + 2cosθ = 1, find the value of 2sinθ – cosθ.

The angle of elevation of the top of a tower 30 m high from the foot of another tower

in the same plane is 60°, and the angle of elevation of the top of second tower from the

foot of first tower is 30°. Find the distance between two towers.

The angle of elevation of the top of a vertical tower from a point on the ground is 60°. From another point 10 m vertically above the first, its angle of elevation is 45°. Find the height of the tower.

State whether the following statment is true or false.

If cosec θ + cot θ = p, then \cos \theta=\frac{p^{2}-1}{p^{2}+1}

State whether the following statment is true or false.

\sqrt{\sec^2\theta+\operatorname{cosec}^2\theta}=\tan\theta+\cot\theta